Cloud radio access network (C-RAN) and mobile edge computing (MEC) have emerged as promising candidates for the next generation access network techniques. Unfortunately, although MEC tries to utilize the highly distributed computing resources in close proximity to user equipments (UE), C-RAN suggests to centralize the baseband processing units (BBU) deployed in radio access networks. To better understand and address such a conflict, this paper closely investigates the MEC task offloading control in C-RAN environments. Most prior work handling offloading control falls in the general category of resource allocation optimization. However in this paper, we focus on the perspective of matching problem. Our model smartly captures the unique features in both MEC and C-RAN with respect to communication and computation efficiency constraints. We divide the cross-layer optimization into the following three stages: (1) matching between remote radio heads (RRH) and UEs, (2) matching between BBUs and UEs, and (3) matching between mobile clones (MC) and UEs. By applying the Gale-Shapley Matching Theory in the duplex matching framework, we propose a multi-stage heuristic to minimize the refusal rate for user's task offloading requests. Trace-based simulation confirms that our solution can successfully achieve near-optimal performance in such a hybrid deployment.